Sunday, June 24, 2012
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The Higgs mass around \(125\GeV\) together with the Higgs vev around \(246\GeV\) and the masses of related particles such as the W-bosons and Z-bosons are much smaller than some other energy scales, e.g. the GUT scale and the Planck scale, where some new interesting effects almost certainly occur.

If we calculate the correction to the Higgs mass from quantum loops of virtual particles whose energy may be very high because we know that physics should make sense up to these very high energy scales, we obtain a quadratically divergent term, too:\[

\delta m_h^2 = \mp C\cdot \Lambda^2.

\] Here, \(\Lambda\) is the maximum energy that particles may carry; of course, you would like to send it to infinity or at least \(m_\text{GUT}\sim 10^{16}\GeV\) or \(m_\text{Planck}\sim 10^{19}\GeV\) except that you can't as it creates havoc in the equations. Dimensional analysis shows that there's enough potential for such terms to arise and unless there is an explanation why they shouldn't arise, Gell-Mann's totalitarian principle pretty much guarantees that they will arise.

So the tree-level Higgs squared mass has to be finely adjusted to a huge number of order \(\pm C\cdot \Lambda^2\) that almost completely (but not quite completely) cancels against the loop corrections that are equally high to yield a result that is as small as experiments suggest – and, incidentally, as small as we need for the existence of stars and life. The probability that the tree-level Higgs mass is so accurately adjusted to lead a tiny leftover after the loop corrections are added is small so theories of this kind apparently predict that our world with a light Higgs boson should be unlikely. That's too bad because if theories predict that the reality is unlikely, they don't look like too good theories. This problem with the "unnaturally low" observed Higgs mass is known as the hierarchy problem.

In April, I have explained that the paragraphs above only represent a technologically shallower way to discuss what the hierarchy problem is. A better way is to discuss a full-fledged theory that is valid up to the very high energy scales. This theory has lots of its own parameters – parameters appropriate for a high-energy scale – and these have to be carefully and unnaturally adjusted for a very light Higgs to emerge.

Recently, we've seen several papers arguing that the hierarchy problem is an illusion. Let me say in advance that I find it plausible that this is what physics will ultimately learn about the hierarchy problem; I just don't think that either of these physicists has actually found the right argument that would show such a thing.

Writing susy in between two dollar signs looks rather stupid but it's not the main problem with the paper. ;-)

Ralston presents examples that demonstrate that the Taylor expansions may be a lousy guide when we want to learn what happens with a function for very high values of the argument. The general claim is right but the examples he chooses have something illogical in them. For example, his example with a Gamma function is one in which the perturbative expansion actually correctly indicates how the function grows for very large values of the argument. It would probably be more logical to choose functions that converge to zero at infinity although the power laws indicate that they diverge.

But I don't want to get into vague interpretations or obvious mathematical tasks. Instead, let's get to the point. The point – the correct point, not Ralston's point – is that we don't have to rely on what Ralston calls the "perturbative" part of the calculation and what sane physicists call "low-energy expansions". Instead, we may and we should directly discuss the models that are valid up to very high energy scales. When we do so, the would-be "non-perturbative" (we actually mean "high-energy-physics-related") corrections are included. And when we do so, an extremely fine adjustment of the parameters of the high-energy theory is needed to yield light particles such as the Higgs boson. That's the hierarchy problem.

We can show that this problem exists in theories that look like ordinary enough theories at the high scale. This proof has no errors in which some vital "non-perturbative" effects would be neglected. We know it's right. The only non-anthropic way to fix the hierarchy problem is to change the theory, to switch to a new theory in which the sensitivity disappears. The known strategies include supersymmetry and its cancellations; new strongly coupled QCD-dynamics at a lower energy scale whose smallness may be explained by the very slow logarithmic running of the coupling; and, less usually, completely new theories resembling the string-theoretical "unexpectedly ultrasoft" short-distance behavior.

But one has to change the laws of physics. The laws of physics have to contain something unusual that actually invalidates the low-energy power-law arguments based on the quadratic divergences. If the theory has no special features, we know from "exact" high-energy calculations that the hierarchy problem is there and that the simple guess based on the quadratic divergences may indeed be extrapolated and produces the right conclusion.

Also, if Ralston's toy model were relevant, you would need not only quadratic but also quartic, sixth-order, and all higher-order divergent terms, too. That would make the theory really pathological and totally non-renormalizable as a quantum field theory although it could be a result of some stringy effects. But almost no one expects these intrinsically stringy effects to kick in below energies close to the GUT scale or Planck scale so it seems unlikely to the physicists that they could have something to do with the solution to the hierarchy problem. There may be a loophole in these expectations but once again, it doesn't seem as though Ralston has found such a loophole.

Starkman et al.

Four days ago, Glenn Starkman wrote a guest blog for the Scientific American whose goal was extremely similar but the strategy was a bit different (or, more precisely, completely different):

He argues that supersymmetry or any other new physics isn't needed as a solution of the hierarchy problem because the problem doesn't exist. He has (and his co-authors have) a different rhetorical strategy to show that the hierarchy problem doesn't exist: the Higgs mass may be linked to the masses of the Goldstone bosons – the angular directions of the Higgs doublet – which have a reason to remain massless, namely the Goldstone theorem.

Except that this claim is wrong at many levels. First of all, the Goldstone bosons don't stay massless in the Standard Model because they're eaten by the W-boson and Z-boson fields and become the gauge bosons' (equally massive) longitudinal polarizations. But even if you considered the spontaneous breaking of a global symmetry and not a local one, it wouldn't work because only the 3 Goldstone bosons – the "angular" components of the Higgs doublet, one for each generator of the symmetry group that is broken – have a reason to remain massless.

So any attempt to claim that the Higgs itself – the radial component – remains light or massless or unaffected by the quadratic divergences is an artifact of circular reasoning. The circular reasoning may have various forms. These guys, for example, decided to describe the system in terms of a low-energy theory based on pions etc. However, if the Higgs isn't made light to start with, the dynamics of this theory isn't described by this approximate low-energy effective theory at all. There aren't any low energies. You can't absorb divergences to the pion masses because the pion masses aren't among the degrees of freedom of the right description.

by Bryan W. Lynn, Glenn D. Starkman, Katherine Freese, Dmitry I. Podolsky. As far as I know, I only know Kathy Freese in person and I only know Dmitry Podolsky from intense communications on the Internet (he has a blog, nonequilibrium.net).

Under Starkman's essay in the Scientific American, I left the following comment.

Exactly one year ago, Dmitry Podolsky, a co-author of yours, had a correspondence – about 10 e-mails – with me. He argued in favor of a simpler “argument” than your paper’s argument why there were no quadratic divergences to the Higgs mass.

The claim was that they only affected the mass parameters so the quartic coupling wasn’t affected by them and the influence of the quadratic divergences on the Higgs mass is actually linked to the correction to the tadpole trying to change the vev (location of the minimum of the potential) which has to be zero for stability. So even the former thing is zero.

For hours if not days, I was a bit confused but then I came back to my senses and wrote him an explanation – one that he never replied to again. The explanation is that the correction of the tadpole is indeed linked to the correction of the Higgs mass and the vev. That’s OK but *none* of these mutually related things is guaranteed to be zero by any principle. In fact, we know in particular theories it’s not zero. The quadratic divergences reflect the big sensitivity of the Higgs mass/vev on all the detailed parameters of any high-energy theory with start with. We may choose to say whether it’s the Higgs mass or the Higgs vev that is threatened by these divergent terms but both of them are!

Without a principle that guarantees a cancellation, both the Higgs vev and the Higgs mass – together with masses of the W-bosons, Z-bosons, and some leptons and quarks – would be close to the GUT scale or any other high-energy scale we find in the full theory. Their fates and values could be correlated but there exists no argument for either of these quantities why they should be small.

Now, the final paper of yours talks about the pions etc. It is a bizarre treatment. From the viewpoint of the fundamental theory, pions are composite objects and their properties are derived quantities. We don’t really have special counterterms for pions or even technipions in the Standard Model. They’re not fundamental parameters. So at most, you link the divergences to yet another quantity that isn’t guaranteed to be zero by any principle. At most, you are supplying a new principle but this principle is equivalent to saying “there shouldn’t be a hierarchy problem”. You don’t have any independent justification why it’s not there.

It’s only the Goldstone bosons that have a reason to be massless, the Goldstone theorem, but there are only 3 of them, one for each broken generator, and the physical Higgs boson simply isn’t one of them. That’s why I believe that your paper boldly claiming that the quadratic divergences aren’t really there is wrong.

So I don't think that either of these authors has addressed the actual hierarchy problem. They have just addressed oversimplified beginners' caricatures of the problem. If they actually understood how the low-energy Higgs physics emerges from some complete theories of the known types that work up to very high energy scales, they would notice how ludicrous their "solutions" are.

We can explicitly see that the fine-tuning of the high-energy-based theory is needed to produce the light particles so hopes inspired by some unusual extrapolations of the incomplete, low-energy approximation of the situation have nothing to do with the reality. In other words, you can't solve the hierarchy problem by looking at it through strong and dirty eyeglasses and by claiming that it doesn't look too sharp in this way.

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Dilaton
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So with my very bad myopia, I probably should not look at the hierarchy problem without using any optical devices ... :-PNice article, I`ve already seen the corresponding question deiscussing the Starkman paper at physics.SE :-)

Assuming then that hierarchy problem does exist, do you (personally) believe the Randall+Sundrum model is a solution of it? After all what is above the cut-off of their effective theory... Would the Higgs not still receive quadratically divergent corrections from some UV physics above the cut-off, such as from string theory?

If so, does that leave SUSY as the only solution? I would be interested to know if you found plausible any alternative solutions to the hierarchy problem.

I read somewhere that the unification of the couplings is predominantly influenced by higgisinos. Is it plausible to postulate that the unification of couplings, a dark matter candidate and a solution to the hierachy problem can be solved with and absolute minimum extension of the standard model with a supersymmetry of only the higgisinos?

Dear Physics Junkie, the reason why higgsinos are important for coupling unification is that leptons and quarks come in complete representations of the grand unified group so they don't affect the unification.

Otherwise no, you can't say what you said because there is no supersymmetry if you cherry-pick particle species one by one. Supersymmetry is a symmetry that - if it exists at all - must constrain the whole spectrum. The right unification of the MSSM is also obtained for higgsinos from two doublets and the fact that one needs two electroweak Higgs doublets (supermultiplets) and not just one has reasons that lie outside coupling unification: it's about the anomaly cancellation and the need to produce masses of both upper and lower quarks.

Moreover, more generally, if one were able to adjust gauge coupling unification by cherry-picking the field content, it wouldn't be impressive at all. The agreement in the MSSM is that the unification is achieved by doing nothing, just postulating SUSY and the minimal content. So even if your "higgsino only" theory would achieve coupling unification, it would be a far less interesting achievement than what one gets in the MSSM.

Hi, I think that RS is a very serious theory but I personally consider the probability that it's relevant in Nature - at accessible energies - to be just 1%-2%.

And no, once you get above the scale where string theory starts to be relevant, there are no quadratic divergences anymore. The dependence on energy gets hugely softer - a typical property of string theory. String theory has no UV divergences, after all.

I am not 100% sure whether SUSY is the right solution to the hierarchy problem. If it is, it should be ultimately be seen at the LHC, kind of. I guess that the probability of either of these statements is about 60%, far below 100% although above 50%.

There have been other proposals to solve the hierarchy problem that seemed worth considering to me. But instead of the technical ones that seem much less well-motivated to me than SUSY (and even than RS), i would mention the anthropic "solution": a light Higgs is simply needed for life so we find outselves in a world where the HIggs is light although there are many more worlds where it is not.

I dislike the anthropic principle but I do admit it's plausible that it will be ultimately needed to explain the lightness of the Higgs. 10% probability or so.

I completely agree with Lubos but want to add an extra comment. Having light gauginos also plays an important role in unification as their contributions to the one-loop beta function coefficients decrease the slopes of the lines, thereby pushing the unification scale from 10^14 to 10^16 GeV, which helps avoid the experimental constraints from the proton decay.

My memory tells me that it was me who made the last reply in that discussion but if I am wrong, I ask you to forgive me. My answer to your observation was more or less as follows.

1. Stability

Whether renormalized tadpole is zero on not shows one whether Higgs particles can be spontaneously produced from the vacuum or not. If they can be produced, then the vacuum is not stable. The particles will be produced from vacuum and the physical Higgs’ VEV will change until the renormalized tadpole becomes zero.

A simple observation which goes back to work done by Bryan Lynn long time ago (and which is actually present in the textbook by Peskin and Schroder citing Bryan’s work) is that the quadratically divergent contribution to the Higgs’ mass coincides identically with the expression for the renormalized tadpole. Hence the first statement of the paper – if renormalized vacuum is stable w.r.t. spontaneous particle production, quadratic divergences are absent in the Higgs’ mass. This observation is actually well known to CMT physicists. There are plenties of phenomena in condensed matter physics involving spontaneously broken global symmetries, and none of them feature quadratic (or linear – because they are (2+1)d) divergences in the effective mass of quasiparticles.

2. Massless modes

I am afraid you cannot simply say that there are several independent generators of global symmetries, so divergences in Goldstone’s masses are unrelated to divergences in Higgs’ mass. Even if the global symmetry is broken spontaneously, effective potential should respect it in some way or another – that’s why the word “spontaneously” (instead of “explicitly”) is used. Whether pions are composite or fundamental objects again does not matter – once you have an effective renormalizable lagrangian for low energy degrees of freedom, you can and should study the fate of divergences appearing in the effective theory.

It so happens that global symmetry relates some contributions to the masses of Goldstones and the Higgs, namely, quadratically divergent contributions. This statement is again known for years and can be found for example in Peskin and Schroder. So, if you want to see that the Goldstone theorem holds explicitly at a given energy scale (i.e, renormalized mass of Goldstones is zero), you should conclude that the quadratically divergent contribution to the Higgs’ mass also vanishes.

I share very much your sentiment about the sensitivity of the Higgs’ mass to parameters of HE theories (that’s why we always thought we need SUSY, Technicolor, etc., etc., etc), but that’s unfortunately not how it works in these theories.

Dear Dmitry, thanks for your comment but I think that you're just repeating the very same observation you sent in the very first e-mail in our conversation, with all the sloppiness, too.

Yes, the renormalization of the Higgs mass is linked to the tadpole - which changes the values of the vev, the point where the vacuum is stable - but *none* of these two related quantities is protected by any principle against huge corrections in the Standard Model. Instead, the huge corrections want to make the Higgs mass very large; and they also want to make the Higgs vev - the expectation value around which we have to expand in order to get a vanishing tadpole - very large. Haven't I explained this in the blog entry above, too?

Concerning your second point, it's just completely wrong because the Higgs mode isn't related to the Goldstone mode by any symmetry - neither in the broken phase nor in the unbroken phase. Goldstone modes are related to each other by the symmetry - they're the angular modes. But the Higgs boson is the radial component and there's simply no symmetry that exchanges e.g. "r" and "phi" in polar coordinates.

Whether you may find "some of the same terms" in two expressions is totally irrelevant. What's important is that this is not true for *all* the terms - which could only be true because of a symmetry or "something equally strong" but there simply isn't anything of the sort.

Note that none of your argument above carries any resemblance to your paper with Starkman and others - you don't talk about any pions etc. in this comment although this is the focus of your paper. So it suggests that even you don't believe the paper you have co-authored. To summarize, there are just several sloppy arguments with clear errors in them. But several sloppy arguments can't replace a valid argument.

My e-mail archive shows that the last message of the conversation was written by me, on June 7th, 2011, and it did find the very same point. I wrote:

Dear Dmitry,I forgot to write that I was silly and I don't think that there is any confusion about your question on the Higgs divergences. (the correction to the Higgs' mass quadratic in cutoff has the same structure as tadpole corrections and the latter are zero by stability of the effective potential)This is a loaded interpretation. While the corrections to the mass are linked to the tadpole, and the tadpole is indeed the same thing as the stability of the effective potential around the same point, NONE of these three things is guaranteed to be zero by any principle. So yes, the quadratic divergences change the Higgs mass and they also change the Higgs vev (the value of the Higgs field at which the potential has a minimum). At any rate, these terms are there and they affect both quantities.